Emulation, Reduction, and Emergence in Dynamical Systems

1. Emulation, Reduction, and Emergence in Dynamical Systems Marco Giunti Università di Cagliari, Italy giunti@unica.it http://edu.supereva.it/giuntihome.dadacasa

2. Outline • The received view about emergence and reduction is that they are incompatible categories. (Beckermann 1992; Kim 1992) • Contrary to the received view, I argue that emergence and reduction can hold together. • In dynamical systems, emulation is sufficient for reduction; • this representational view of reduction, contrary to the standarddeductivist one, is compatible with the existence of structural properties of the reduced system that are not also properties of the reducing one. Thus, under this view, reduction and emergence are not incompatible.

3. A classic definition of emergence • Intuitively, a property of a high level system is said to be emergent if it is not one of the properties of more basic parts, which, together, make up the system. More precisely: • A property P of a high level system S2isemergent with respect to a lower level system S1 just in case (a) S2 is made up of S1 (intuitively, S1 is the system of the constitutive parts of S2 taken in isolation, or in relations different from those typical of S2; see Broad 1925) and (b) P is not one of the properties of S1.

4. Reduction – received view • It is obvious that, if S2 is reduced to S1, then all properties of S2 are properties of S1; • thus, by condition (b) of the definition of emergence, it follows that emergence and reduction are incompatible. • My view: by no means is the above principle obvious; in fact, it is false. It thus follows that emergence and reduction may hold together.

5. The argumentative strategy • To support this thesis, I will focus attention on dynamical systems, and on the emulation relationship between them; • in virtue of a general representation theorem, I will argue that, for any two dynamical systems, the emulation relationship is sufficient for both reduction andconstitution (i.e., the being made up of relationship); • therefore, to show that both reduction and emergence can hold together, it will suffice to exhibit two dynamical systems DS1 and DS2, as well as a property P, such that DS1 emulates DS2,DS2 has P, but DS1 does not have P.

6. Example of a continuous Dyn. Syst. The Galilean model of free fall • Explicit specification Let F = (M, (gt)tÎT) such that • M = S´V and S = V = T = real numbers • gt(s, v) = (s + vt + at2/2, v + at) • Implicit specification Let F = (M, (gt)tÎT) such that • M = S´V and S = V = T = real numbers • ds(t)/dt = v(t), dv(t)/dt = a

7. Example of a discrete Dyn. Syst.A finite Cellular Automaton Twelve cells arranged in a circle. The value of each cell is either 0 or 1. Thus, the CA has 212 = 4096 possible states. Rule 111 110 101 100 011 010 001 000 0 1 0 1 1 0 1 0 Rule number 010110102 = 9010 Time 0 0 1 0 0 1 0 1 0 1 1 1 1 Time 1 0 0 1 1 0 0 0 0 1 0 0 1

8. A Dynamical System (DS) is a mathematical model that expresses the idea of a deterministic system (discrete/continuous, revers./irrevers.) • A Dynamical System (DS) is a set theoretical structure (M, (gt)tÎT) such that: • the set M is not empty; M is called the state-space of the system; • the set T, is either Z, Z+ (integers) or R, R+ (reals); T is called the time set; • (gt)tÎT is a family of functions from M to M; each function gt is called a state transition or a t-advance of the system; • for any t and wÎ T, for any xÎ M, • g0(x) = x; • gt+w(x) = gw(gt(x)).

9. Intuitive meaning of the definition of dynamical system gt gt(x) x t t0 t0+t g0 gt+w gw x gt x

10. Isomorphism between two DSs Definition uis anisomorphism ofDS2 = (N, (hv)vV)inDS1 = (M, (gt)tT)iff T = V, u: N  M is a bijection and, for any vV, for any cN, u(hv(c)) = gv(u(c)). DS2is isomorphic toDS1iff there is u which is anisomorphism of DS2in DS1 M N u gv hv c u

11. Emulation between two DSs Intuition and examples • Intuitively, a DSemulates a second DSwhen the first one exactly reproduces the whole dynamics of the second one. • Examples (i) a universal Turing machine emulates all TMs; (ii) for any TM there is a cellular automaton CA that emulates TM, and vice versa; (iii) emulation holds between two binary CAs with neighborhood of radius 1 (Wolfram’s rule 22 emulates rule 146).

12. Emulation between two DSs Definition uis anemulation of DS2= (N, (hv)vV)inDS1= (M, (gt)tT)iff u: N  M is an injection and, for any vV, for any cN, there is tT such that u(hv(c)) = gt(u(c)) DS1emulatesDS2iff there is u which is anemulation of DS2in DS1 M N u(N) u gt hv c u

13. Virtual System Theorem [VST] If u is an emulation ofDS2 = (N, (hv)vV) in DS1 = (M, (gt)tT), there is a third system DS3 = (N, (hv)vV) such that (i)uis an isomorphism of DS2 in DS3; (ii) all states of DS3 are states of DS1[because N = u(N)] ; (iii) any state transition hv of DS3 is constructed out of state transitions of DS1. M N u(N) u hv gt gt hv a c u u-1 DS3 is called the virtualu-system DS2 inDS1

14. Emulation →constitution and reduction • Because of [VST], if a dynamical system DS1 emulates a second system DS2, it makes perfect sense to claim that DS2is made up ofDS1, as well as that DS2is reduced toDS1. • In other words, I maintain that, in virtue of [VST],emulation is sufficient for bothconstitutionandreduction.

15. M N h1 c z u h1 g1 g1 h1 y u b g1 u x a Emergence and reduction hold together: example • a pair of cascadesDS1 = (M, (gt)tZ+) and DS2 = (N, (hv)vZ+) such that (i)DS2is reduced toDS1 and (ii) the property P of strong irreversibility is an emergent property of DS2 with respect to DS1 • DS1emulatesDS2, DS1is logically reversible (thus, not strongly irreversible), and DS2is strongly irreversible

16. That’s all Thank you

17. References Beckermann, Ansgar (1992), “Supervenience, Emergence and Reduction”, in Ansgar Beckermann, Tommaso Toffoli, and Jaegwon Kim (eds.), Emergence or Reduction?Essays on the Prospects of Nonreductive Physicalism. Berlin: Walter de Gruyter, 94-118. Broad, Charlie Dunbar (1925), The Mind and its Place in Nature. London: Routledge and Kegan Paul. Kim, Jaegwon (1992), “Downward Causation in Emergentism and Non-reductive Physicalism”, in Ansgar Beckermann, Tommaso Toffoli, and Jaegwon Kim (eds.), Emergence or Reduction?Essays on the Prospects of Nonreductive Physicalism. Berlin: Walter de Gruyter, 119-138.